First passage percolation on random graphs with finite mean
degrees
Citation for published version (APA):
Bhamidi, S., Hofstad, van der, R. W., & Hooghiemstra, G. (2009). First passage percolation on random graphs with finite mean degrees. (Report Eurandom; Vol. 2009063). Eurandom.
Document status and date: Published: 01/01/2009
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arXiv:0903.5136v2 [math.PR] 13 Oct 2009
First passage percolation on random graphs with finite mean degrees
Shankar Bhamidi ∗ Remco van der Hofstad † Gerard Hooghiemstra‡ October 13, 2009
Abstract
We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount.
We analyze the configuration model with degree power-law exponent τ > 2, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of power-law form with exponent τ− 1 > 1, or has even thinner tails (τ = ∞). In this model, the degrees have a finite first moment, while the variance is finite for τ > 3, but infinite for τ ∈ (2, 3).
We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to α log n, where α∈ (0, 1) for τ ∈ (2, 3), while α > 1 for τ > 3. Here n denotes the size of the graph. For τ ∈ (2, 3), it is known that the graph distance between two randomly chosen connected vertices is proportional to log log n [25], i.e., distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path, and prove convergence in distribution of an appropriately centered version.
This study continues the program initiated in [5] of showing that log n is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees (τ ∈ [1, 2)) is studied in [6], where it is proved that the hopcount remains uniformly bounded and converges in distribution.
Key words: Flows, random graph, first passage percolation, hopcount, central limit theorem, coupling to continuous-time branching processes, universality.
MSC2000 subject classification. 60C05, 05C80, 90B15.
1
Introduction
The general study of real-world networks has seen a tremendous growth in the last few years. This growth occurred both at an empirical level of obtaining data on networks such as the Internet, transportation networks, such as rail and road networks, and biochemical networks, such as gene regulatory networks, as well as at a theoretical level in the understanding of the properties of various mathematical models for these networks.
We are interested in one specific theoretical aspect of the above vast and expanding field. The setting is as follows: Consider a transportation network whose main aim is to transport flow between various vertices in the network via the available edges. At the very basic level there are two crucial elements which affect the flow carrying capabilities and delays experienced by vertices in the network:
∗Department of Statistics and Operations Research, 304 Hanes Hall, University of North Carolina, Chapel Hill, NC.
E-mail: bhamidi@email.unc.edu
†Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB
Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl
‡EEMCS, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands. E-mail:
(a) The actual graph topology, such as the density of edges and existence of short paths between vertices in the graph distance. In this context there has been an enormous amount of interest in the concept of small-world networks where the typical graph distance between vertices in the network is of order log n or even smaller. Indeed, for many of the mathematical models used to model real-world transmission networks, such as the Internet, the graph distance can be of order much smaller than order
log n. See e.g. [13, 25], where for the configuration model with degree exponent τ ∈ (2, 3), the remarkable
result that the graph distance between typical vertices is of order log log n is proved. In this case, we say that the graph is ultra small, a phrase invented in [13]. Similar results have appeared for related models in [11, 16, 33]. The configuration model is described in more detail in Section 2. For introductions to scale-free random graphs, we refer to the monographs [12, 17], for surveys of classical random graphs focussing on the Erd˝os-R´enyi random graph, see [8, 30].
(b) The second factor which plays a crucial role is the edge weight or cost structure of the graph, which can be thought of as representing actual economic costs or congestion costs across edges. Edge weights being identically equal to 1 gives us back the graph geometry. What can be said when the edge costs have some other behavior? The main aim of this study is to understand what happens when each edge is given an independent edge cost with mean 1. For simplicity, we have assumed that the distribution of edge costs is exponentially with mean 1 (Exp(1)), leading to first passage percolation on the graph involved. First passage percolation with exponential weights has received substantial attention (see [5, 22, 23, 26, 27, 37]), in particular on the complete graph, and, more recently, also on Erd˝os-R´enyi random graphs. However, particularly the relation to the scale-free nature of the underlying random graph and the behavior of first passage percolation on it has not yet been investigated.
In this paper, we envisage a situation where the edge weights represent actual economic costs, so that all flow is routed through minimal weight paths. The actual time delay experienced by vertices in
the network is given by the number of edges on this least cost path or hopcount Hn. Thus, for two
typical vertices 1 and 2 in the network, it is important to understand both the minimum weight Wn of
transporting flow between two vertices as well as the hopcount Hnor the number of edges on this minimal
weight path. What we shall see is the following universal behavior:
Even if the graph topology is of ultra-small nature, the addition of random edge weights causes a complete change in the geometry and, in particular, the number of edges on the minimal weight path between two vertices increases to Θ(log n).
Here we write an = Θ(bn) if there exist positive constants c and C, such that, for all n, we have cbn ≤
an ≤ Cbn. For the precise mathematical results we refer to Section 3. We shall see that a remarkably
universal picture emerges, in the sense that for each τ > 2, the hopcount satisfies a central limit theorem
(CLT) with asymptotically equal mean and variance equal to α log n, where α ∈ (0, 1) for τ ∈ (2, 3),
while α > 1 for τ > 3. The parameter α is the only feature which is left from the randomness of the
underlying random graph, and α is a simple function of τ for τ ∈ (2, 3), and of the average forward degree
for τ > 3. This type of universality is reminiscent of that of simple random walk, which, appropriately scaled, converges to Brownian motion, and the parameters needed for the Brownian limit are only the mean and variance of the step-size. Interestingly, for the Internet hopcount, measurements show that the hopcount is close to a normal distribution with equal mean and variance (see e.g., [36]), and it would be of interest to investigate whether first passage percolation on a random graph can be used as a model for the Internet hopcount.
This paper is part of the program initiated in [5] to rigorously analyze the asymptotics of distances and weights of shortest-weigh paths in random graph models under the addition of edge weights. In this paper, we rigorously analyze the case of the configuration model with degree exponent τ > 2, the conceptually important case in practice, since the degree exponent of a wide variety of real-world networks
is conjectured to be in this interval. In [6], we investigate the case τ ∈ [1, 2), where the first moment of
2
Notation and definitions
We are interested in constructing a random graph on n vertices. Given a degree sequence, namely a
sequence of n positive integers d = (d1, d2, . . . , dn) with Pni=1di assumed to be even, the configuration
model (CM) on n vertices with degree sequence d is constructed as follows:
Start with n vertices and di stubs or half-edges adjacent to vertex i. The graph is constructed by
randomly pairing each stub to some other stub to form edges. Let
ln=
n
X
i=1
di (2.1)
denote the total degree. Number the stubs from 1 to ln in some arbitrary order. Then, at each step, two
stubs which are not already paired are chosen uniformly at random among all the unpaired or free stubs and are paired to form a single edge in the graph. These stubs are no longer free and removed from the list of free stubs. We continue with this procedure of choosing and pairing two stubs until all the stubs are paired. Observe that the order in which we choose the stubs does not matter. Although self-loops
may occur, these become rare as n→ ∞ (see e.g. [8] or [28] for more precise results in this direction).
Above, we have described the construction of the CM when the degree sequence is given. Here we shall specify how we construct the actual degree sequence d, which shall be random. In general, we
shall let a capital letter (such as Di) denote a random variable, while a lower case letter (such as di)
denote a deterministic object. We shall assume that the random variables D1, D2, . . . Dnare independent
and identically distributed (i.i.d.) with a certain distribution function F . (When the sum of stubs
Ln = Pni=1Di is not even then we shall use the degree sequence D1, D2, . . . , Dn, with Dn replaced by
Dn+ 1. This does not effect our calculations.)
We shall assume that the degrees of all vertices are at least 2 and that the degree distribution F is regularly varying. More precisely, we assume
P(D≥ 2) = 1, and 1− F (x) = x−(τ −1)L(x), (2.2)
with τ > 2, and where x 7→ L(x) is a slowly varying function for x → ∞. In the case τ > 3, we
shall replace (2.2) by the less stringent condition (3.2). Furthermore, each edge is given a random edge weight, which in this study will always be assumed to be independent and identically distributed (i.i.d.) exponential random variables with mean 1. Because in our setting the vertices are exchangeable, we let 1 and 2 be the two random vertices picked uniformly at random in the network.
As stated earlier, the parameter τ is assumed to satisfy τ > 2, so that the degree distribution has
finite mean. In some cases, we shall distinguish between τ > 3 and τ ∈ (2, 3), in the former case, the
variance of the degrees is finite, while in the latter, it is infinite. It follows from the condition Di ≥ 2,
almost surely, that the probability that the vertices 1 and 2 are connected converges to 1.
Let f ={fj}∞j=1 denote the probability mass function corresponding to the distribution function F ,
so that fj = F (j)− F (j − 1). Let {gj}∞j=1 denote the size-biased probability mass function corresponding
to f , defined by
gj =
(j + 1)fj+1
µ , j≥ 0, (2.3)
where µ is the expected size of the degree, i.e.,
µ = E[D] =
∞
X
j=1
3
Results
In this section, we state the main results for τ > 2. We treat the case where τ > 3 in Section 3.1 and the
case where τ ∈ (2, 3) in Section 3.2. The case where τ ∈ [1, 2) is deferred to [6].
Throughout the paper, we shall denote by
(Hn, Wn), (3.1)
the number of edges and total weight of the shortest-weight path between vertices 1 and 2 in the CM with i.i.d. degrees with distribution function F , where we condition the vertices 1 and 2 to be connected and we assume that each edge in the CM has an i.i.d. exponential weight with mean 1.
3.1 Shortest-weight paths for τ > 3
In this section, we shall assume that the distribution function F of the degrees in the CM is non-degenerate
and satisfies F (x) = 0, x < 2, so that the random variable D is non-degenerate and satisfies D ≥ 2, a.s.,
and that there exist c > 0 and τ > 3 such that
1− F (x) ≤ cx−(τ −1), x≥ 0. (3.2)
Also, we let
ν = E[D(D− 1)]
E[D] . (3.3)
As a consequence of the conditions we have that ν > 1. The condition ν > 1 is equivalent to the existence of a giant component in the CM, the size of which is proportional to n (see e.g. [24, 31, 32], for the most recent and general result, see [29]). Moreover, the proportionality constant is the survival probability
of the branching process with offspring distribution {gj}j≥1. As a consequence of the conditions on the
distribution function F , in our case, the survival probability equals 1, so that for n → ∞ the graph
becomes asymptotically connected in the sense that the giant component has n(1− o(1)) vertices. Also,
when (3.2) holds, we have that ν < ∞. Throughout the paper, we shall let −→ denote convergence ind
distribution and−→ convergence in probability.P
Theorem 3.1 (Precise asymptotics for τ > 3) Let the degree distribution F of the CM on n vertices be non-degenerate, satisfy F (x) = 0, x < 2, and satisfy (3.2) for some τ > 3. Then,
(a) the hopcount Hn satisfies the CLT
Hn− α log n
√ α log n
d
−→ Z, (3.4)
where Z has a standard normal distribution, and
α = ν
ν− 1∈ (1, ∞); (3.5)
(b) there exists a random variable V such that
Wn−
log n
ν− 1
d
−→ V. (3.6)
In Section C of the appendix, we shall identify the limiting random variable V as
V =−log W1 ν− 1 − log W2 ν− 1 + Λ ν− 1+ log µ(ν− 1) ν− 1 , (3.7)
where W1, W2 are two independent copies of the limiting random variable of a certain supercritical
3.2 Analysis of shortest-weight paths for τ ∈ (2, 3)
In this section, we shall assume that (2.2) holds for some τ ∈ (2, 3) and some slowly varying function
x7→ L(x). When this is the case, the variance of the degrees is infinite, while the mean degree is finite. As
a result, we have that ν in (3.3) equals ν =∞, so that the CM is always supercritical (see [25, 29, 31, 32]).
In fact, for τ ∈ (2, 3), we shall make a stronger assumption on F than (2.2), namely, that there exists a
τ ∈ (2, 3) and 0 < c1 ≤ c2 <∞ such that, for all x ≥ 0,
c1x−(τ −1)≤ 1 − F (x) ≤ c2x−(τ −1). (3.8)
Theorem 3.2 (Precise asymptotics for τ ∈ (2, 3)) Let the degree distribution F of the CM on n
ver-tices be non-degenerate, satisfy F (x) = 0, x < 2, and satisfy (3.8) for some τ ∈ (2, 3). Then,
(a) the hopcount Hn satisfies the CLT
Hn− α log n
√ α log n
d
−→ Z, (3.9)
where Z has a standard normal distribution and where
α = 2(τ− 2)
τ − 1 ∈ (0, 1); (3.10)
(b) there exists a limiting random variable V such that
Wn−→ V.d (3.11)
In Section 6, we shall identify the limiting distribution V precisely as
V = V1+ V2, (3.12)
where V1, V2 are two independent copies of a random variable which is the explosion time of a certain
infinite-mean continuous-time branching process.
3.3 Discussion and related literature
Motivation. The basic motivation of this work was to show that even though the underlying graph
topology might imply that the distance between two vertices is very small, if there are edge weights representing capacities, say, then the hopcount could drastically increase. Of course, the assumption of i.i.d. edge weights is not very realistic, however, it allows us to almost completely analyze the minimum weight path. The assumption of exponentially distributed edge weights is probably not necessary [1, 27] but helps in considerably simplifying the analysis. Interestingly, hopcounts which are close to normal with asymptotically equal means and variances are observed in Internet (see e.g., [36]). The results presented here might shed some light on the origin of this observation.
Universality for first passage percolation on the CM. Comparing Theorem 3.1 and Theorem 3.2
we see that a remarkably universal picture emerges. Indeed, the hopcount in both cases satisfies a CLT
with equal mean and variance proportional to log n, and the proportionality constant α satisfies α∈ (0, 1)
for τ ∈ (2, 3), while α > 1 for τ > 3. We shall see that the proofs of Theorems 3.1 and 3.2 run, to a large
extent, parallel, and we shall only need to distinguish when dealing with the related branching process problem to which the neighborhoods can be coupled.
The case τ ∈ [1, 2) and critical cases τ = 2 and τ = 3. In [6], we study first passage percolation on
the CM when τ ∈ [1, 2), i.e., the degrees have infinite mean. We show that a remarkably different picture
emerges, in the sense that Hn remains uniformly bounded and converges in distribution. This is due to
the fact that we can think of the CM, when τ ∈ [1, 2), as a union of an (essentially) finite number of stars.
Together with the results in Theorems 3.1–3.2, we see that only the critical cases τ = 2 and τ = 3 remain open. We conjecture that the CLT, with asymptotically equal means and variances, remains valid when
τ = 3, but that the proportionality constant α can take any value in [1,∞), depending on, for example,
whether ν in (3.3) is finite or not. What happens for τ = 2 is less clear to us.
Graph distances in the CM. Expanding neighborhood techniques for random graphs have been used
extensively to explore shortest path structures and other properties of locally tree-like graphs. See the closely related papers [19, 24, 25, 34], where an extensive study of the CM has been carried out. Relevant to our context is [25, Corollary 1.4(i)], where it has been shown that when 2 < τ < 3, the graph distance
e
Hn between two typical vertices,which are conditioned to be connected, satisfies the asymptotics
e Hn log log n P −→ 2 | log (τ − 2)|, (3.13)
as n→ ∞, and furthermore that the fluctuations of eHnremain uniformly bounded as n→ ∞. For τ > 3,
it is shown in [24, Corollary 1.3(i)] that
e Hn log n P −→ 1 log ν, (3.14)
again with bounded fluctuations. Comparing these results with Theorems 3.1–3.2, we see the drastic effect that the addition of edge weights has on the geometry of the graph.
The degree structure. In this paper, as in [19, 24, 25, 34], we assume that the degrees are i.i.d. with
a certain degree distribution function F . In the literature, also the setting where the degrees {di}ni=1 are
deterministic and converge in an appropriate sense to an asymptotic degree distribution is studied (see e.g., [11, 20, 29, 31, 32]). We expect that our results can be adapted to this situation. Also, we assume that the degrees are at least 2 a.s., which ensures that two uniform vertices lie, with high probability (whp) in the giant component. We have chosen for this setting to keep the proofs as simple as possible, and we conjecture that Theorems 3.1–3.2, when instead we condition the vertices 1 and 2 to be connected, remain true verbatim in the more general case of the supercritical CM.
Annealed vs. quenched asymptotics. The problem studied in this paper, first passage percolation
on a random graph, fits in the more general framework of stochastic processes in random environments, such as random walk in random environment. In such problems, there are two interesting settings, namely, when we study results when averaging out over the environment and when we freeze the environment (the so-called annealed and quenched asymptotics). In this paper, we study the annealed setting, and it would be of interest to extend our results to the quenched setting, i.e., study the first-passage percolation problem conditionally on the random graph. We expect the results to change in this case, primarily due to the fact that we know the exact neighborhood of each point. However, when we consider the shortest-weight problem between two uniform vertices, we conjecture Theorems 3.1–3.2 to remain valid verbatim, due to the fact that the neighborhoods of uniform vertices converge to the same limit as in the annealed setting (see e.g., [4, 24]).
First passage percolation on the Erd˝os-R´enyi random graph. We recall that the Erd˝os-R´enyi
random graph G(n, p) is obtained by taking the vertex set [n] = {1, . . . , n} and letting each edge ij be
where similar ideas were explored for dense Erd˝os-R´enyi random graphs. The Erd˝os-R´enyi random graph
G(n, p) can be viewed as a close brother of the CM, with Poisson degrees, hence with τ =∞. Consider
the case where p = µ/n and µ > 1. In a future paper we plan to show, parallel to the above analysis, that
Hn satisfies a CLT with asymptotically equal mean and variance given by µ−1µ log n. This connects up
nicely with [5], where related results were shown for µ = µn→ ∞, and Hn/ log n was proved to converge
to 1 in probability. See also [22] where related statements were proved under stronger assumptions on
µn. Interestingly, in a recent paper, Ding et al [15] use first passage percolation to study the diameter of
the largest component of the Erd˝os-R´enyi random graph with edge probability p = (1 + ε)/n for ε = o(1)
and ε3n→ ∞.
The weight distribution. It would be of interest to study the effect of weights even further, for
example, by studying the case where the weights are i.i.d. random variables with distribution equal to
Es, where E is an exponential random variable with mean 1 and s∈ [0, ∞). The case s = 0 corresponds
to the graph distance ˜Hn as studied in [19, 24, 25], while the case s = 1 corresponds to the case with i.i.d.
exponential weights as studied here. Even the problem on the complete graph seems to be open in this case, and we intend to return to this problem in a future paper. We conjecture that the CLT remains
valid for first passage perolation on the CM when the weights are given by independent copies of Es, with
asymptotic mean and variance proportional to log n, but, when s 6= 1, we predict that the asymptotic
means and variances have different constants.
We became interested in random graphs with edge weights from [9] where, via empirical simulations, a wide variety of behavior was predicted for the shortest-weight paths in various random graph models. The setup that we analyze is the weak disorder case. In [9], also a number of interesting conjectures regarding the strong disorder case were made, which would correspond to analyzing the minimal spanning tree of these random graph models, and which is a highly interesting problem.
Related literature on shortest-weight problems. First passage percolation, especially on the
in-teger lattice, has been extensively studied in the last fifty years, see e.g. [18] and the more recent survey [26]. In these papers, of course, the emphasis is completely different, in the sense that geometry plays an intrinsic role and often the goal of the study is to show that there is a limiting “shape” to first passage percolation from the origin.
Janson [27] studies first passage percolation on the complete graph, with exponential weights. His main results are
W(ij) n log n/n P −→ 1, maxj≤nW (ij) n log n/n P −→ 2, maxi,j≤nW (ij) n log n/n P −→ 3. (3.15) where W(ij)
n denotes the weight of the shortest path between the vertices i and j. Recently the authors
of [1], showed in the same set-up, that maxi,j≤nHn(ij)/ log n −→ αP ⋆, where α⋆ ≈ 3.5911 is the unique
solution of the equation x log x− x = 1. It would be of interest to investigate such questions in the CM
with exponential weights.
The fundamental difference of first passage percolation on the integer lattice, or even on the complete graph, is that in our case the underlying graph is random as well, and we are lead to the delicate relation between the randomness of the graph together with that of the stochastic process, in this case first passage percolation, living on it. Finally, for a slightly different perspective to shortest weight problems, see [37] where relations between the random assignment problem and the shortest-weight problem with exponential edge weights on the complete graph are explored.
4
Overview of the proof and organization of the paper
The key idea of the proof is to first grow the shortest-weight graph (SWG) from vertex 1, until it reaches an appropriate size. After this, we grow the SWG from vertex 2 until it connects up with the SWG from
vertex 1. The size to which we let the SWG from 1 grow shall be the same as the typical size at which the connection between the SWG from vertices 1 and 2 shall be made. However, the connection time at which the SWG from vertex 2 connects to the SWG from vertex 1 is random.
More precisely, we define the SWG from vertex 1, denoted by SWG(1), recursively. The growth of
the SWG from vertex 2, which is denoted by SWG(2), is similar. We start with vertex 1 by defining
SWG(1)
0 ={1}. Then we add the edge and vertex with minimal edge weight connecting vertex 1 to one
of its neighbors (or itself when the minimal edge is a self-loop). This defines SWG(1)1 . We obtain SWG(1)
m
from SWG(1)
m−1 by adding the edge and end vertex connected to the SWGm−1 with minimal edge weight.
We informally let SWG(i)
m denote the SWG from vertex i∈ {1, 2} when m edges (and vertices) have been
added to it. This definition is informal, as we shall need to deal with self-loops and cycles in a proper way.
How we do this is explained in more detail in Section 4.2. As mentioned before, we first grow SWG(1)
m
to a size an, which is to be chosen appropriately. After this, we grow SWG(2)m, and we stop as soon as a
vertex of SWG(1)
an appears in{SWG
(2)
m}∞m=0, as then the shortest-weight path between vertices 1 and 2 has
been found. Indeed, if on the contrary, the shortest weight path between vertex 1 and vertex 2 contains an edge not contained in the union of the two SWGs when they meet, then necessarily this edge would have been chosen in one of the two SWGs at an earlier stage, since at some earlier stage this edge must have been incident to one of the SWGs and had the minimal weight of all edges incident to that SWG. In Sections 4.2 and 4.3, we shall make these definitions precise.
Denote this first common vertex by A, and let Gi be the distance between vertex i and A, i.e., the
number of edges on the minimum weight path from i to A. Then, we have that
Hn= G1+ G2, (4.1)
while, denoting by Ti the weight of the shortest-weight paths from i to A, we have
Wn= T1+ T2. (4.2)
Thus, to understand the random variables Hnand Wn, it is paramount to understand the random variables
Ti and Gi, for i = 1, 2.
Since, for n→ ∞, the topologies of the neighborhoods of vertices 1 and 2 are close to being
indepen-dent, it seems likely that G1 and G2, as well as T1 and T2 are close to independent. Since, further, the
CM is locally tree-like, we are lead to the study of the problem on a tree. With the above in mind, the paper is organized as follows:
• In Section 4.1 we study the flow on a tree. More precisely, in Proposition 4.3, we describe the asymptotic distribution of the length and weight of the shortest-weight path between the root and
the mthadded vertex in a branching process with i.i.d. degrees with offspring distribution g in (2.3).
Clearly, the CM has cycles and self-loops, and, thus, sometimes deviates from the tree description. • In Section 4.2, we reformulate the problem of the growth of the SWG from a fixed vertex as a problem of the SWG on a tree, where we find a way to deal with cycles by a coupling argument, so that the arguments in Section 4.1 apply quite literally. In Proposition 4.6, we describe the asymptotic distribution of the length and weight of the shortest-weight path between a fixed vertex
and the mth added vertex in the SWG from the CM. However, observe that the random variables
Gi described above are the generation of a vertex at the time at which the two SWGs collide, and
this time is a random variable.
• In Section 4.3, we extend the discussion to this setting, and formulate the necessary ingredients for the collision time, i.e., the time at which the connecting edge appears, in Proposition 4.4. In Section 4.5, we complete the outline.
• The proofs of the key propositions are deferred to Sections 5–7.
• Technical results needed in the proofs in Sections 5–7, for example on the topology of the CM, are deferred to the appendix.
4.1 Description of the flow clusters in trees
We shall now describe the construction of the SWG in the context of trees. In particular, below, we shall deal with a flow on a branching process tree, where the offspring is deterministic.
Deterministic construction: Suppose we have positive (non-random) integers d1, d2, . . .. Consider the
following construction of a branching process in discrete time:
Construction 4.1 (Flow from root of tree) The shortest-weight graph on a tree with degrees{di}∞i=1
is obtained as follows:
1. At time 0, start with one alive vertex (the initial ancestor);
2. At each time step i, pick one of the alive vertices at random, this vertex dies giving birth to di
children.
In the above construction, the number of offspring diis fixed once and for all. For a branching process
tree, the variables di are i.i.d. random variables. This case shall be investigated later on, but the case of
deterministic degrees is more general, and shall be important for us to be able to deal with the CM. Consider a continuous-time branching process defined as follows:
1. Start with the root which dies immediately giving rise to d1 alive offspring;
2. Each alive offspring lives for Exp(1) amount of time, independent of all other randomness involved;
3. When the mth vertex dies it leaves behind dm alive offspring.
The split-times (or death-times) of this branching process are denoted by Ti, i ≥ 1. Note that
the Construction 4.1 is equivalent to this continuous branching process, observed at the discrete times
Ti, i≥ 1. The fact that the chosen alive vertex is chosen at random follows from the memoryless property
of the exponential random variables that compete to become the minimal one.) We quote a fundamental result from [10]. In its statement, we let
si = d1+· · · + di− (i − 1).1 (4.3)
Proposition 4.2 (Shortest-weight paths on a tree) Pick an alive vertex at time m ≥ 1 uniformly
at random among all vertices alive at this time. Then,
(a) the generation of the mth chosen vertex is equal in distribution to
Gm=d
m
X
i=1
Ii, (4.4)
where {Ii}∞i=1 are independent Bernoulli random variables with
P(Ii = 1) = di/si, (4.5)
(b) the weight of the shortest-weight path between the root of the tree and the vertex chosen in the mth
step is equal in distribution to
Tm =d
m
X
i=1
Ei/si, (4.6)
where {Ei}∞i=1 are i.i.d. exponential random variables with mean 1.
1A new probabilistic proof is added, since there is some confusion between the definition s
igiven here, and the definition of sigiven in [10, below Equation (3.1)]. More precisely, in [10], siis defined as si= d1+ . . . + di−i, which is our si− 1.
Proof. We shall prove part (a) by induction. The statement is trivial for m = 1. We next assume that
(4.4) holds for m, where {Ii}mi=1 are independent Bernoulli random variables satisfying (4.5). Let Gm+1
denote the generation of the randomly chosen vertex at time m + 1, and consider the event {Gm+1 =
k}, 1 ≤ k ≤ m. If randomly choosing one of the alive vertices at time m + 1 results in one of the dm+1
newly added vertices, then, in order to obtain generation k, the previous uniform choice, i.e., the choice
of the vertex which was the last one to die, must have been a vertex from generation k− 1. On the other
hand, if a uniform pick is conditioned on not taking one of the dm+1 newly added vertices, then this
choice must have been a uniform vertex from generation k. Hence, we obtain, for 1≤ k ≤ m,
P(Gm+1 = k) = dm+1 sm+1 P(Gm= k− 1) + 1−dm+1 sm+1 P(Gm = k). (4.7)
The proof of part (a) is now immediate from the induction hypothesis. The proof of part(b) is as follows.
The minimum of si independent exp(1) random variables has an exponential distribution with parameter
si, and is hence equal in distribution to Ei/si. We further use the memoryless property of the exponential
distribution which guarantees that at each of the discrete time steps the remaining lifetimes (or weights) of the alive vertices are independent exponential variables with mean 1, independent of what happened previously.
We note that, while Proposition 4.2 was applied in [10, Theorem 3.1] only in the case where the degrees are i.i.d., in fact, the results hold more generally for every tree (see e.g., [10, Equation (3.1)], and the above proof). This extension shall prove to be vital in our analysis.
We next intuitively relate the above result to our setting. Start from vertex 1, and iteratively choose the edge with minimal additional weight attached to the SWG so far. As mentioned before, because of the properties of the exponential distribution, the edge with minimal additional weight can be considered to be picked uniformly at random from all edges attached to the SWG at that moment. With high
probability, this edge is connected to a vertex which is not in the SWG. Let Bi denote the forward degree
(i.e., the degree minus 1) of the vertex to which the ith edge is connected. By the results in [24, 25],
{Bi}i≥2 are close to being i.i.d., and have distribution given by (2.3). Therefore, we are lead to studying
random variables of the form (4.4)–(4.5), where {Bi}∞i=1 are i.i.d. random variables. Thus, this means
that we study the unconditional law of Gm in (4.4), in the setting where the vector {di}∞i=1 is replaced
by an i.i.d. sequence of random variables {Bi}∞i=1. We shall first state a CLT for Gm and a limit result
for Tm in this setting. In its statement, we shall also make use of the random variable eTm, which is the
weight of the shortest weight path between the root and the parent of the mthindividual in the branching
process. Thus, in particular, eTm ≤ Tm, and Tm− eTm is the time between the addition of the parent of
the mth individual and the mth individual itself.
Proposition 4.3 (Asymptotics for shortest-weight paths on trees) Let {Bi}∞i=1 be an i.i.d.
se-quence of non-degenerate, positive integer valued, random variables, satisfying
P(Bi > k) = k2−τL(k), τ > 2,
for some slowly varying function k 7→ L(k). Denote by ν = E[B1], for τ > 3, whereas ν = ∞, for
τ ∈ (2, 3). Then,
(a) for Gm given in (4.4)–(4.5), with di replaced by Bi, there exists a β≥ 1 such that, as m → ∞,
Gm− β log m
√
β log m
d
−→ Z, where Z ∼ N (0, 1), (4.8)
a standard normal variable, and where β = ν/(ν− 1) for τ > 3, while β = 1 for τ ∈ (2, 3);
(b) for Tm given in (4.6), there exists random variables X, eX such that
Tm− γ log m−→ X,d Tem− γ log m−→ ed X, (4.9)
Proposition 4.3 is proved in [10, Theorem 3.1] when Var(Bi) < ∞, which holds when τ > 4, but not
when τ ∈ (2, 4). We shall prove Proposition 4.3 in Section 5 below. There, we shall also see that the result
persists under weaker assumptions than{Bi}∞i=1being i.i.d., for example, when{Bi}∞i=1are exchangeable
non-negative integer valued random variables satisfying certain conditions. Such extensions shall prove to be useful when dealing with the actual (forward) degrees in the CM.
4.2 A comparison of the flow on the CM and the flow on the tree
Proposition 4.3 gives a CLT for the generation when considering a flow on a tree. In this section, we shall relate the problem of the flow on the CM to the flow on a tree. The key feature of this construction is that we shall simultaneously grow the graph topology neighborhood of a vertex, as well as the shortest-weight graph from it. This will be achieved by combining the construction of the CM as described in Section 2 with the fact that, from a given set of vertices and edges, if we grow the shortest-weight graph, each potential edge is equally likely to be the minimal one.
In the problem of finding the shortest weight path between two vertices 1 and 2, we shall grow two SWGs simultaneously from the two vertices 1 and 2, until they meet. This is the problem that we actually need to resolve in order to prove our main results in Theorems 3.1-3.2. The extension to the growth of two SWGs is treated in Section 4.3 below.
The main difference between the flow on a graph and on a tree is that on the tree there are no cycles, while on a graph there are. Thus, we shall adapt the growth of the SWG for the CM in such a way that we obtain a tree (so that the results from Section 4.1 apply), while we can still retrieve all information about shortest-weight paths from the constructed graph. This will be achieved by introducing the notion of artificial vertices and stubs. We start by introducing some notation.
We denote by {SWGm}m≥0 the SWG process from vertex 1. We construct this process recursively.
We let SWG0 consist only of the alive vertex 1, and we let S0 = 1. We next let SWG1 consist of the D1
allowed stubs and of the explored vertex 1, and we let S1= S0+D1−1 = D1 denote the number of allowed
stubs. In the sequel of the construction, the allowed stubs correspond to vertices in the shortest-weight
problem on the tree in Section 4.1. This constructs SWG1. Next, we describe how to construct SWGm
from SWGm−1. For this construction, we shall have to deal with several types of stubs:
(a) the allowed stubs at time m, denoted by ASm, are the stubs that are incident to vertices of the SWGm,
and that have not yet been paired to form an edge; Sm=|ASm| denotes their number;
(b) the free stubs at time m, denoted by FSm, are those stubs of the Ln total stubs which have not yet
been paired in the construction of the CM up to and including time m;
(c) the artificial stubs at time m, denoted by Artm, are the artificial stubs created by breaking ties, as
described in more detail below.
We note that Artm⊂ ASm, indeed, ASm\FSm = Artm. Then, we can construct SWGmfrom SWGm−1
as follows. We choose one of the Sm−1 allowed stubs uniformly at random, and then, if the stub is not
artificial, pair it uniformly at random to a free stub unequal to itself. Below, we shall consistently call these two stubs the chosen stub and the paired stub, respectively. There are 3 possibilities, depending on what kind of stub we choose and what kind of stub it is paired to:
Construction 4.4 (The evolution of SWG for CM as SWG on a tree)
(1) The chosen stub is real, i.e., not artificial, and the paired stub is not one of the allowed stubs. In this case, which shall be most likely at the start of the growth procedure of the SWG, the paired stub is
incident to a vertex outside SWGm−1, we denote by Bm the forward degree of the vertex incident to the
paired stub (i.e, its degree minus 1), and we define Sm = Sm−1 + Bm− 1. Then, we remove the paired
and the chosen stub from ASm−1 and add the Bm stubs incident to the vertex incident to the paired stub
to ASm−1 to obtain ASm, we remove the chosen and the paired stubs from FSm−1 to obtain FSm, and
Artm = Artm−1;
to a vertex in SWGm−1 and we have created a cycle. In this case, we create an artificial stub replacing
the paired stub, and denote Bm = 0. Then, we let Sm = Sm−1− 1, remove both the chosen and paired
stubs from ASm−1 and add the artificial stub to obtain ASm, and remove the chosen and paired stub from
FSm−1 to obtain FSm, while Artm is Artm−1 together with the newly created artificial stub. In SWGm,
we also add an artificial edge to an artificial vertex in the place where the chosen stub was, the forward degree of the artificial vertex being 0. This is done because a vertex is added each time in the construction on a tree.
(3) The chosen stub is artificial. In this case, we let Bm = 0, Sm = Sm−1 − 1, and remove the chosen
stub from ASm−1 and Artm−1 to obtain ASm and Artm, while FSm = FSm−1.
In the construction in Construction 4.4, we always work on a tree since we replace an edge which creates a cycle, by one artificial stub, to replace the paired stub, and an artificial edge plus an artificial
vertex in the SWGm with degree 0, to replace the chosen stub. Note that the number of allowed edges
at time m satisfies Sm= Sm−1+ Bm− 1, where B1= D1 and, for m ≥ 2, in cases (2) and (3), Bm = 0,
while in case (1) (which we expect to occur in most cases), the distribution of Bm is equal to the forward
degree of a vertex incident to a uniformly chosen stub. Here, the choice of stubs is without replacement. The reason for replacing cycles as described above is that we wish to represent the SWG problem as
a problem on a tree, as we now will explain informally. On a tree with degrees {di}∞i=1, as in Section
4.1, we have that the remaining degree of vertex i at time m is precisely equal to di minus the number
of neighbors that are among the m vertices with minimal shortest-weight paths from the root. For first passage percolation on a graph with cycles, a cycle does not only remove one of the edges of the vertex incident to it (as on the tree), but also one edge of the vertex at the other end of the cycle. Thus, this is a different problem, and the results from Section 4.1 do not apply literally. By adding the artificial stub, edge and vertex, we artificially keep the degree of the receiving vertex the same, so that we do have the same situation as on a tree, and we can use the results in Section 4.1. However, we do need to investigate the relation between the problem with the artificial stubs and the original SWG problem on the CM. That is the content of the next proposition.
In its statement, we shall define the mth closest vertex to vertex 1 in the CM, with i.i.d. exponential
weights, as the unique vertex of which the minimal weight path is the mth smallest among all n − 1
vertices. Further, at each time m, we denote by artificial vertices those vertices which are artificially
created, and we call the other vertices real vertices. Then, we let the random time Rm be the first time
j that SWGj consists of m + 1 real vertices, i.e.,
Rm= minj≥ 0 : SWGj contains m + 1 real vertices . (4.10)
The +1 originates from the fact that at time m = 0, SWG0 consists if 1 real vertex, namely, the vertex
from which we construct the SWG. Thus, in the above set up, we have that Rm = m precisely when no
cycle has been created in the construction up to time m. Then, our main coupling result is as follows:
Proposition 4.5 (Coupling shortest-weight graphs on a tree and CM) Jointly for all m ≥ 1,
the set of real vertices in SWGRm is equal in distribution to the set of ith closest vertices to vertex
1, for i = 1, . . . , m. Consequently,
(a) the generation of the mth closest vertex to vertex 1 has distribution G
Rm, where Gm is defined in
(4.4)–(4.5) with d1= D1 and di = Bi, i≥ 2, as described in Construction 4.4;
(b) the weight of the shortest weight path to the mth closest vertex to vertex 1 has distribution TRm, where
Tm is defined in (4.6) with d1 = D1 and di = Bi, i≥ 2, as described in Construction 4.4.
We shall make use of the nice property that the sequence {BRm}
n
m=2, which consists of the forward
degrees of chosen stubs that are paired to stubs which are not in the SWG, is, for the CM, an exchangeable sequence of random variables (see Lemma 6.1 below). This is due to the fact that a free stub is chosen uniformly at random, and the order of the choices does not matter. This exchangeability shall prove to be useful in order to investigate shortest-weight paths in the CM. We now prove Proposition 4.5:
Proof of Proposition 4.5. In growing the SWG, we give exponential weights to the set {ASm}m≥1.
After pairing, we identify the exponential weight of the chosen stub to the exponential weight of the edge which it is part of. We note that by the memoryless property of the exponential random variable, each stub is chosen uniformly at random from all the allowed stubs incident to the SWG at the given time. Further, by the construction of the CM in Section 2, this stub is paired uniformly at random to one of the available free stubs. Thus, the growth rules of the SWG in Construction 4.4 equal those in the
above description of{SWGm}∞m=0, unless when a cycle is closed and an artificial stub, edge and vertex are
created. In this case, the artificial stub, edge and vertex might influence the law of the SWG. However, we note that the artificial vertices are not being counted in the set of real vertices, and since artificial vertices have forward degree 0, they will not be a part of any shortest path to a real vertex. Thus, the artificial vertex at the end of the artificial edge does not affect the law of the SWG. Artificial stubs that are created to replace paired stubs when a cycle is formed, and which are not yet removed at time m, will be called dangling ends. Now, if we only consider real vertices, then the distribution of weights and lengths of the shortest-weight paths between the starting points and those real vertices are identical. Indeed, we can decorate any graph with as many dangling ends as we like without changing the shortest-weight paths to real vertices in the graph.
Now that the flow problem on the CM has been translated into a flow problem on a related tree of which we have explicitly described its distribution, we may make use of Proposition 4.2, which shall allow us to extend Proposition 4.3 to the setting of the CM. Note that, among others due to the fact that when we draw an artificial stub, the degrees are not i.i.d. (and not even exchangeable since the probability of drawing an artificial stub is likely to increase in time), we need to extend Proposition 4.3 to a setting
where the degrees are weakly dependent. In the statement of the result, we recall that Gm is the height
of the mth added vertex in the tree problem above. In the statement below, we write
an= n(τ ∧3−2)/(τ ∧3−1) =
(
n(τ −2)/(τ −1) for τ ∈ (2, 3);
n1/2 for τ > 3, (4.11)
where, for a, b∈ R, we write a ∧ b = min{a, b}.
Before we formulate the CLT for the hopcount of the shortest-weight graph in the CM, we repeat
once more the setup of the random variables involved. Let S0 = 1, S1 = D1, and for j≥ 2,
Sj = D1+
j
X
i=2
(Bi− 1), (4.12)
where, in case the chosen stub is real, i.e., not artificial, and the paired stub is not one of the allowed
stubs, Biequals the forward degree of the vertex incident to the ithpaired stub, whereas Bi = 0 otherwise.
Finally, we recall that, conditionally on D1, B2, B3, . . . , Bm,
Gm =
m
X
i=1
Ii, where P(I1= 1) = 1, P(Ij = 1) = Bj/Sj, 2≤ j ≤ m. (4.13)
Proposition 4.6 (Asymptotics for shortest-weight paths in the CM) (a) Let the law of Gm be
given in (4.13). Then, with β ≥ 1 as in Proposition 4.3, and as long as m ≤ mn, for any mn such that
log (mn/an) = o(√log n), Gm− β log m √ β log m d −→ Z, where Z ∼ N (0, 1). (4.14)
(b) Let the law of Tm be given in (4.6), with sireplaced by Si given by (4.12) and let γ be as in Proposition
4.3. Then, there exists a random variable X such that
The same results apply to GRm and TRm, i.e., in the statements (a) and (b) the integer m can be replaced
by Rm, as long as m≤ mn.
Proposition 4.6 implies that the result of Proposition 4.3 remains true for the CM, whenever m is not too large. Important for the proof of Proposition 4.6 is the coupling to a tree problem in Proposition 4.5. Proposition 4.6 shall be proved in Section 6. An important ingredient in the proof will be the comparison
of the variables {Bm}mm=2n , for an appropriately chosen mn, to an i.i.d. sequence. Results in this direction
have been proved in [24, 25], and we shall combine these to the following statement:
Proposition 4.7 (Coupling the forward degrees to an independent sequence) In the CM with
τ > 2, there exists a ρ > 0 such that the random vector {Bm}n
ρ
m=2 can be coupled to an independent
sequence of random variables {B(ind)
m }n
ρ
m=2 with probability mass function g in (2.3) in such a way that
{Bm}n ρ m=2 ={B (ind) m }n ρ m=2 whp.
In Proposition 4.7, in fact, we can take {Bm}n
ρ
m=2 to be the forward degree of the vertex to which any
collection of nρdistinct stubs has been connected.
4.3 Flow clusters started from two vertices
To compute the hopcount, we first grow the SWG from vertex 1 until time an, followed by the growth
of the SWG from vertex 2 until the two SWGs meet, as we now explain in more detail. Denote by
{SWG(i)
m}∞m=0 the SWG from the vertex i∈ {1, 2}, and, for m ≥ 0, let
SWG(1,2)
m = SWG(1)an∪ SWG
(2)
m, (4.16)
the union of the SWGs of vertex 1 and 2. We shall only consider values of m where SWG(1)
an and SWG
(2)
m
are disjoint, i.e., they do not contain any common (real) vertices. We shall discuss the moment when they connect in Section 4.4 below.
We recall the notation in Section 4.2, and, for i∈ {1, 2}, denote by AS(i)
m and Art(i)m the number of
allowed and artificial stubs in SWG(i)
m. We let the set of free stubs FSm consist of those stubs which
have not yet been paired in SWG(1,2)
m in (4.16). Apart from that, the evolution of SWG(2)m, following the
evolution of SWG(1)
an, is identical as in Construction 4.4. We denote by S
(i)
m =|AS(i)m| the number of allowed
stubs in SWG(i)
m for i∈ {1, 2}. We define B
(i)
m accordingly.
The above description shows how we can grow the SWG from vertex 1 followed by the one of vertex 2. In order to state an adaptation of Proposition 4.5 to the setting where the SWGs of vertex 1 is first
grown to size an, followed by the growth of the SWG from vertex 2 until the connecting edge appears,
we let the random time R(i)m be the first time l such that SWG(i)l consists of m + 1 real vertices. Then,
our main coupling result for two simultaneous SWGs is as follows:
Proposition 4.8 (Coupling SWGs on two trees and CM from two vertices) Jointly for m≥ 0,
as long as the sets of real vertices in (SWG(1)
an, SWG
(2)
m) are disjoint, these sets are equal in distribution
to the sets of j1th, respectively j2th, closest vertices to vertex 1 and 2, respectively, for j1= 1, . . . , Ra(1)n and
j2 = 1, . . . , R(2)m, respectively.
4.4 The connecting edge
As described above, we grow the two SWGs until the first stub with minimal weight incident to SWG(2)
m
is paired to a stub incident to SWG(1)
an. We call the created edge linking the two SWGs the connecting
edge. More precisely, let
Cn= min{m ≥ 0 : SWG(1)an∩ SWG
(2)
m 6= ∅}, (4.17)
be the first time that SWG(1)
an and SWG
(2)
m share a vertex. When m = 0, this means that 2 ∈ SWG(1)an
of SWG(2) which is chosen and then paired, is paired to a stub from SWG(1)
an. The path found actually
is the shortest-weight path between vertices 1 and 2, since SWG(1)
an and SWG
(2)
m precisely consists of the
closest real vertices to the root i, for i = 1, 2, respectively.
We now study the probabilistic properties of the connecting edge. Let the edge e = st be incident to
SWG(1)
an, and s and t denote its two stubs. Let the vertex incident to s be is and the vertex incident to t
be it. Assume that is ∈ SWG(1)an, so that, by construction, it 6∈ SWG
(1)
an. Then, conditionally on SWG
(1)
an
and {T(1)
i }ai=1n , the weight of e is at least T
(1)
an − W (1)
is , where W
(1)
is is the weight of the shortest path from 1
to is. By the memoryless property of the exponential distribution, therefore, the weight on edge e equals
T(1)
an − W (1)
is + Ee, where the collection (Ee), for all e incident to SWG
(1)
an are i.i.d. Exp(1) random variables.
Alternatively, we can redistribute the weight by saying that the stub t has weight Ee, and the stub s has
weight T(1)
an − W (1)
is . Further, in the growth of (SWG
(2)
m)m≥0, we can also think of the exponential weights
of the edges incident to SWG(2)
m being positioned on the stubs incident to SWG(2)m. Hence, there is no
distinction between the stubs that are part of edges connecting SWG(1)
an and SWG
(2)
m and the stubs that
are part of edges incident to SWG(2)
m, but not to SWG(1)an. Therefore, in the growth of (SWG
(2)
m)m≥0, we
can think of the minimal weight stub incident to SWG(2)
m being chosen uniformly at random, and then a
uniform free stub is chosen to pair it with. As a result, the distribution of the stubs chosen at the time of connection is equal to any of the other (real) stubs chosen along the way. This is a crucial ingredient to prove the scaling of the shortest-weight path between vertices 1 and 2.
For i∈ {1, 2}, let H(i)
n denote the length of the shortest-weight path between vertex i and the common
vertex in SWG(1)
an and SWG
(2)
Cn, so that
Hn= Hn(1)+ Hn(2). (4.18)
Because of the fact that at time Cn we have found the shortest-weight path, we have that
(H(1) n , Hn(2)) d = (G(1) an+1− 1, G (2) Cn), (4.19) where {G(1) m}∞m=1 and {G (2)
m}∞m=1 copies of the process in (4.4), which are conditioned on drawing a real
stub. Indeed, at the time of the connecting edge, a uniform (real) stub of SWG(2)
m is drawn, and it is
paired to a uniform (real) stub of SWG(1)
an. The number of hops in SWG
(1)
an to the end of the attached edge
is therefore equal in distribution to G(1)
an+1 conditioned on drawing a real stub. The−1 in (4.19) arises
since the connecting edge is counted twice in G(1)an+1+ G(2)Cn. The processes {G
(1)
m}∞m=1 and {G
(2)
m}∞m=1 are
conditionally independent given the realizations of {B(i)
m}nm=2.
Further, because of the way the weight of the potential connecting edges has been distributed over the two stubs out of which the connecting edge is comprised, we have that
Wn= Ta(1)n + T (2) Cn, (4.20) where {T(1) m }∞m=1 and {T (2)
m }∞m=1 are two copies of the process {Tm}∞m=1 in (4.6), again conditioned on
drawing a real stub. Indeed, to see (4.20), we note that the weight of the connecting edge is equal to the sum of weights of its two stubs. Therefore, the weight of the shortest weight path is equal to the sum of
the weight within SWG(1)
an, which is equal to T
(1)
an, and the weight within SWG
(2)
Cn, which is equal to T
(2)
Cn.
In the distributions in (4.19) and (4.20) above, we always condition on drawing a real stub. Since we shall show that this occurs whp, this conditioning plays a minor role.
We shall now intuitively explain why the leading order asymptotics of Cn is given by an, where an
is defined in (4.11). For this, we must know how many allowed stubs there are, i.e., we must determine
how many stubs there are incident to the union of the two SWGs at any time. Recall that S(i)
m denotes
the number of allowed stubs in the SWG from vertex i at time m. The total number of allowed stubs
incident to SWG(1)
an is S (1)
an, while the number incident to SWG
(2) m is equal to S (2) m, and where S(i) m = Di+ m X l=2 (B(i) l − 1). (4.21)
We also write Artm = Art(1)an∪ Art (2) m. Conditionally on SWG(1) an and{(S (2) l , Art (2) l )} m−1
l=1 and Ln, and assuming that|Artm|, m and Sm satisfy
appropriate bounds, we obtain
P(Cn= m|Cn> m− 1) ≈ S
(1)
an
Ln
. (4.22)
When τ ∈ (2, 3) and (3.8) holds, then S(i)
l /l1/(τ −2) can be expected to converge in distribution to a
stable random variable with parameter τ − 2, while, for τ > 3, S(i)
l /l converges in probability to ν− 1,
where ν is defined in (3.3). We can combine these two statements by saying that S(i)
l /l1/(τ ∧3−2) converges
in distribution. Note that the typical size anof Cnis such that, uniformly in n, P(Cn∈ [an, 2an]) remains
in (ε, 1− ε), for some ε ∈ (0,1
2), which is the case when
P(Cn∈ [an, 2an]) =
2an
X
m=an
P(Cn= m|Cn> m− 1)P(Cn > m− 1) ∈ (ε, 1 − ε), (4.23)
uniformly as n→ ∞. By the above discussion, and for an≤ m ≤ 2an, we have P(Cn= m|Cn> m− 1) =
Θ(m1/(τ ∧3−2)/n) = Θ(a1/(τ ∧3−2)n /n), and P(Cn> m− 1) = Θ(1). Then, we arrive at
P(Cn∈ [an, 2an]) = Θ(ana1/(τ ∧3−2)n /n), (4.24)
which remains uniformly positive and bounded for an defined in (4.11). In turn, this suggests that
Cn/an−→ M,d (4.25)
for some limiting random variable M .
We now discuss what happens when (2.2) holds for some τ ∈ (2, 3), but (3.8) fails. In this case,
there exists a slowly varying function n 7→ ℓ(n) such that S(i)
l /(ℓ(l)l1/(τ −2)) converges in distribution.
Then, following the above argument shows that the right-hand side (r.h.s.) of (4.24) is replaced by Θ(ana1/(τ −2)n ℓ(an)/n), which remains uniformly positive and bounded for ansatisfying a(τ −1)/(τ −2)n ℓ(an) =
n. By [7, Theorem 1.5.12], there exists a solution an to the above equation which satisfies that it is
regularly varying with exponent (τ − 2)/(τ − 1), so that
an= n(τ −2)/(τ −1)ℓ∗(n), (4.26)
for some slowly varying function n7→ ℓ∗(n), which depends only on the distribution function F .
In the following proposition, we shall state the necessary result on Cn that we shall need in the
remainder of the proof. In its statement, we shall use the symbol oP(bn) to denote a random variable Xn
which satisfies that Xn/bn−→ 0.P
Proposition 4.9 (The time to connection) As n → ∞, under the conditions of Theorems 3.2 and
3.1 respectively, and with an as in (4.11),
log Cn− log an= oP(
p
log n). (4.27)
Furthermore, for i∈ {1, 2}, and with β ≥ 1 as in Proposition 4.3,
G(1)an+1− β log an √ β log an ,G (2) C√n− β log an β log an d −→ (Z1, Z2), (4.28)
where Z1, Z2 are two independent standard normal random variables. Moreover, with γ as in Proposition
4.3, there exist random variables X1, X2 such that
T(1) an − γ log an, T (2) Cn− γ log an d −→ (X1, X2). (4.29)
We note that the main result in (4.28) is not a simple consequence of (4.27) and Proposition 4.6.
The reason is that Cn is a random variable, which a priori depends on (G(1)an+1, G
(2)
m) for m≥ 0. Indeed,
the connecting edge is formed out of two stubs which are not artificial, and thus the choice of stubs is not completely uniform. However, since there are only few artificial stubs, we can extend the proof of Proposition 4.6 to this case. Proposition 4.9 shall be proved in Section 7.
4.5 The completion of the proof
By the analysis in Section 4.4, we know the distribution of the sizes of the SWGs at the time when the connecting edge appears. By Proposition 4.9, we know the number of edges and their weights used in the paths leading to the two vertices of the connecting edge, together with its fluctuations. In the final step, we need to combine these results by averaging both over the randomness of the time when the connecting edge appears (which is a random variable), as well as over the number of edges in the shortest weight path when we know the time the connecting edge appears. Note that by (4.19) and Proposition 4.9, we
have, with Z1, Z2 denoting independent standard normal random variables, and with Z = (Z1+ Z2)/
√ 2, which is again standard normal,
Hn= Gd (1)an+1+ G (2) Cn− 1 = 2β log an+ Z1 p β log an+ Z2 p β log an+ oP( p log n) = 2β log an+ Z p 2β log an+ oP( p log n). (4.30)
Finally, by (4.11), this gives (3.4) and (3.9) with
α = lim
n→∞
2β log an
log n , (4.31)
which equals α = ν/(ν− 1), when τ > 3, since β = ν/(ν − 1) and log an
log n = 1/2, and α = 2(τ− 2)/(τ − 1),
when τ ∈ (2, 3), since β = 1 and log an
log n = (τ− 2)/(τ − 1). This completes the proof for the hopcount.
In the description of α in (4.31), we note that when ancontains a slowly varying function for τ ∈ (2, 3)
as in (4.26), then the result in Theorem 3.2 remains valid with α log n replaced by
2 log an=
2(τ− 2)
τ − 1 log n + 2 log ℓ
∗(n). (4.32)
For the weight of the minimal path, we make use of (4.20) and (4.29) to obtain in a similar way that
Wn− 2γ log an−→ Xd 1+ X2. (4.33)
This completes the proof for the weight of the shortest path.
5
Proof of Proposition 4.3
5.1 Proof of Proposition 4.3(a)
We start by proving the statement for τ ∈ (2, 3). Observe that, in this context, di = Bi, and, by (4.3),
B1+ . . . + Bi= Si+ i− 1, so that the sequence Bj/(Si+ i− 1), for j satisfying 1 ≤ j ≤ i, is exchangeable
for each i≥ 1. Therefore, we define
ˆ Gm = m X i=1 ˆ Ii, where P( ˆIi = 1|{Bi}∞i=1) = Bi Si+ i− 1 . (5.1)
Thus, ˆIiis, conditionally on{Bi}∞i=1, stochastically dominated by Ii, for each i, which, since the sequences
stochastically dominated by Gm. We take ˆGm and Gm in such a way that ˆGm ≤ Gm a.s. Then, by the
Markov inequality, for κm > 0,
P(|Gm− ˆGm| ≥ κm) ≤ κ−1m E[|Gm− ˆGm|] = κ−1m E[Gm− ˆGm] = κ−1m m X i=1 E Bi(i− 1) Si(Si+ i− 1) ] = κ−1m m X i=1 i− 1 i E[1/Si], (5.2)
where, in the second equality, we used the exchangeability of Bj/(Si + i− 1), 1 ≤ j ≤ i. We will now
show that
∞
X
i=1
E[1/Si] <∞, (5.3)
so that for any κm → ∞, we have that P(|Gm − ˆGm| ≤ κm) → 1. We can then conclude that the
CLT for Gm follows from the one for ˆGm. By [14, (3.12) for s = 1], for τ ∈ (2, 3) and using that
Si = B1+· · · Bi− (i − 1), where P(B1 > k) = k2−τL(k), there exists a slowly varying function i7→ l(i)
such that E[1/Si]≤ cl(i)i−1/(τ −2). When τ ∈ (2, 3), we have that 1/(τ − 2) > 1, so that (5.3) follows.
We now turn to the CLT for ˆGm, Observe from the exchangeability of Bj/(Si+ i− 1), for 1 ≤ j ≤ i,
that for i1 < i2 < . . . < ik, P( ˆIi1 = . . . = ˆIik = 1) = E hYk l=1 Bil Sil+ il− 1 i = Eh Bi1 Si1+ i1− 1 k Y l=2 Bil Sil+ il− 1 i = 1 i1 Eh k Y l=2 Bil Sil+ il− 1 i = . . . = k Y l=1 1 il , (5.4)
where we used that since B1+ . . . + Bj = Sj+ j− 1,
Eh Bi1 Si1 + i1− 1 k Y l=2 Bil Sil+ il− 1 i = 1 i1 i1 X i=1 Eh Bi Si1+ i1− 1 k Y l=2 Bil Sil+ il− 1 i = 1 i1 Eh k Y l=2 Bil Sil+ il− 1 i .
Since ˆIi1, . . . , ˆIik are indicators this implies that ˆIi1, . . . , ˆIik are independent. Thus, ˆGm has the same
distribution as Pmi=1Ji, where {Ji}∞i=1 are independent Bernoulli random variables with P(Ji = 1) =
1/i. It is a standard consequence of the Lindeberg-L´evy-Feller CLT that (Pmi=1Ji − log m)/√log m is
asymptotically standard normally distributed.
Remark 5.1 (Extension to exchangeable setting) Note that the CLT for Gm remains valid when
(i) the random variables{Bi}mi=1are exchangeable, with the same marginal distribution as in the i.i.d. case,
and (ii) Pmi=1E[1/Si] = o(√log m).
The approach for τ > 3 is different from that of τ ∈ (2, 3). For τ ∈ (2, 3), we coupled Gm to ˆGm and
proved that ˆGmsatisfies the CLT with the correct norming constants. For τ > 3, the case we consider now,
we first apply a conditional CLT, using the Lindeberg-L´evy-Feller condition, stating that, conditionally
on B1, B2, . . . satisfying lim m→∞ m X j=1 Bj Sj 1−Bj Sj =∞, (5.5) we have that Gm−Pmj=1Bj/Sj Pm j=1 Bj Sj(1− Bj Sj) 1/2 d −→ Z, (5.6)
where Z is standard normal. The result (5.6) is also contained in [10].
Since ν = E[Bj] > 1 and E[Bja] <∞, for any a < τ − 2, it is not hard to see that the random variable
P∞
j=1Bj2/S2j is positive and has finite first moment, so that for m→ ∞,
m
X
j=1
B2j/Sj2= OP(1), (5.7)
where OP(bm) denotes a sequence of random variables Xm for which|Xm|/bm is tight.
We claim that m X j=1 Bj/Sj− ν ν− 1log m = oP( p log m). (5.8)
Obviously, (5.6), (5.7) and (5.8) imply Proposition 4.3(a) when τ > 3. In order to prove (5.8), we split
m X j=1 Bj/Sj − ν ν− 1log m = Xm j=1 (Bj− 1)/Sj− log m + m X j=1 1/Sj− 1 ν− 1log m , (5.9)
and shall prove that each of these two terms on the r.h.s. of (5.9) is oP(
√
log m). For the first term, we note from the strong law of large numbers that
m X j=1 log Sj Sj−1
= log Sm− log S0= log m + OP(1). (5.10)
Also, since − log (1 − x) = x + O(x2), we have that
m X j=1 log Sj/Sj−1=− m X j=1 log 1− (Bj − 1)/Sj= m X j=1 (Bj− 1)/Sj + O Xm j=1 (Bj− 1)2/Sj2 . (5.11) Again, as in (5.7), for m→ ∞, m X j=1 (Bj− 1)2/Sj2= OP(1), (5.12) so that m X j=1 (Bj − 1)/Sj− log m = OP(1). (5.13)
In order to study the second term on the right side of (5.9), we shall prove a slightly stronger result than necessary, since we shall also use this later on. Indeed, we shall show that there exists a random variable Y such that m X j=1 1/Sj− 1 ν− 1log m a.s. −→ Y. (5.14)
In fact, the proof of (5.14) is a consequence of [3, Theorem 1], since E[(Bi− 1) log(Bi− 1)] < ∞ for τ > 3.
We decided to give a separate proof of (5.14) which can be easily adapted to the exchangeable case. To prove (5.14), we write m X j=1 1/Sj− 1 ν− 1log m = m X j=1 (ν− 1)j − Sj Sj(ν− 1)j + OP(1), (5.15)
so that in order to prove (5.14), it suffices to prove that, uniformly in m≥ 1, m X j=1 |Sj − (ν − 1)j| Sj(ν− 1)j <∞, a.s. (5.16)
Thus, if we further make use of the fact that Sj ≥ ηj except for at most finitely many j (see also Lemma
A.4 below), then we obtain that m X j=1 1 Sj − 1 ν− 1log m ≤ m X j=1 |Sj− (ν − 1)j| Sj(ν− 1)j + OP(1)≤ C m X j=1 |Sj∗| j2 , (5.17) where S∗
j = Sj− E[Sj], since E[Sj] = (ν− 1)j + 1. We now take the expectation, and conclude that for
any a > 1, Jensen’s inequality for the convex function x7→ xa, yields
E[|Sj∗|] ≤ E[|Sj∗|a]1/a. (5.18)
To bound the last expectation, we will use a consequence of the Marcinkiewicz-Zygmund inequality,
see e.g. [21, Corollary 8.2 on p. 152]. Taking 1 < a < τ − 2, we have that E[|B1|a] <∞, since τ > 3, so
that Eh m X j=1 |S∗ j| j2 i ≤ m X j=1 E[|S∗ j|a]1/a j2 ≤ m X j=1 c1/aa E[|B1|a]1/a j2−1/a <∞. (5.19)
This completes the proof of (5.14).
Remark 5.2 (Discussion of exchangeable setting) When the random variables{Bi}mi=1are
exchange-able, with the same marginal distribution as in the i.i.d. case, and with τ > 3, we note that to prove a
CLT for Gm, it suffices to prove (5.7) and (5.8). The proof of (5.8) contains two steps, namely, (5.13)
and (5.16). For the CLT to hold, we in fact only need that the involved quatities are oP(
√
log m), rather
than OP(1). For this, we note that
(a) the argument to prove (5.13) is rather flexible, and shows that if (i) log Sm/m = oP(
√
log m) and if
(ii) the condition in (5.7) is satisfied with OP(1) replaced by oP(
√
log m), then (5.13) follows with OP(1)
replaced by oP(
√ log m);
(b) for the proof of (5.16) we will make use of stochastic domination and show that each of the stochastic
bounds will satisfy (5.16) with OP(1) replaced by oP(
√
log m) (compare Lemma A.8).
5.2 Proof of Proposition 4.3(b)
We again start by proving the result for τ ∈ (2, 3). It follows from (4.6) and the independence of {Ei}i≥1
and {Si}i≥1 that, for the proof of (4.9), it is sufficient to show that
∞
X
i=1
E[1/Si] <∞, (5.20)
which holds due to (5.3). The argument for eTm is similar, with the same limit. Indeed, on the one hand,
since eTm ≤ Tm, the limit of eTm cannot be larger than that of Tm. On the other hand, since Gm → ∞
and m7→ Tm is increasing, we have that the eTm≥ Tk, whp for any k. Therefore, eTm must have the same
limit as Tm.
The extension of this result to τ > 3, where the weak limits of Tm and eTm are different, is deferred to
